We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field Q of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type E6 . We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.